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Exploring the Actinide-Actinide Bond: Theoretical Studies of
the Chemical Bond in Ac2, Th2, Pa2, and U2
Björn O. Roos,*,† Per-A° ke Malmqvist,† and Laura Gagliardi‡
Contribution from the Department of Theoretical Chemistry, UniVersity of Lund, Chemical
Center, PO Box 124, S-221 00 Lund, Sweden, and Department of Physical Chemistry, Sciences
II, UniVersity of GeneVa, 30 Quai Ernest Ansermet, CH-1211 GeneVa 4, Switzerland
Received September 26, 2006; E-mail: [email protected]
Abstract: Multiconfigurational quantum chemical methods (CASSCF/CASPT2) have been used to study
the chemical bond in the actinide diatoms Ac2, Th2, Pa2, and U2. Scalar relativistic effects and spin-orbit
coupling have been included in the calculations. In the Ac2 and Th2 diatoms the atomic 6d, 7s, and 7p
orbitals are the significant contributors to the bond, while for the two heavier diatoms, the 5f orbitals become
increasingly important. Ac2 is characterized by a double bond with a 3∑g-(0g+) ground state, a bond distance
of 3.64. Å, and a bond energy of 1.19 eV. Th2 has quadruple bond character with a 3Dg(1g) ground state.
The bond distance is 2.76 Å and the bond energy (D0) 3.28 eV. Pa2 is characterized by a quintuple bond
with a 3∑g-(0g+) ground state. The bond distance is 2.37 Å and the bond energy 4.00 eV. The uranium
diatom has also a quintuple bond with a 7Og (8g) ground state, a bond distance of 2.43 Å, and a bond
energy of 1.15 eV. It is concluded that the strongest bound actinide diatom is Pa2, characterized by a
well-developed quintuple bond.
1 Introduction
We have recently demonstrated that multiconfigurational
quantum chemistry can be used to explore the chemical bond
that is formed between two uranium atoms,1 and also for the
corresponding dipositive ion, U22+.2 Other earlier studies of
possible multiple bonds between two uranium ions include
molecules such as PhUUPh3 and a number of chlorides and
formates.4 Here we give a more detailed account of the U2 study
and also add results from similar calculations of the diatoms of
the three earlier actinides, Ac, Th, and Pa.
Not much is known experimentally about the chemical bond
between actinide atoms. The uranium molecule has been
detected in gas phase, but it has never been isolated.5 The
dissociation energy was estimated to be 52(5) kcal/mol, a value
which must be considered as highly uncertain. U2 and U2+ have
also been seen using laser vaporization of a solid uranium
target.6 Of the other diactinides Th2 has been detected both in
gas phase6 and in a rare gas matrix,7 but no properties are yet
known of either of these two molecules.
The study of the uranium diatom showed a very complex
electronic structure. A quintuple bond connects the two atoms.
†
‡
University of Lund.
University of Geneva.
(1) Gagliardi, L.; Roos, B. O. Nature 2005, 433, 848.
(2) Gagliardi, L.; Pyykkö, P.; Roos, B. O. Phys. Chem. Chem. Phys. 2005, 7,
2415.
(3) Macchia, G. L.; Brynda, M.; Gagliardi, L. Angew. Chem., Int. Ed. 2006,
45, 6210.
(4) Roos, B. O.; Gagliardi, L. Inorg. Chem. 2006, 45, 803.
(5) Gorokhov, L. N.; Emelyanov, A. M.; Khodeev, Y. S. Teplofiz. Vys. Temp.
1974, 12, 1307.
(6) Heaven, M. C. Private communication.
(7) Andrews, L. Private communication.
17000
9
J. AM. CHEM. SOC. 2006, 128, 17000-17006
It is built from three electron pair bonds formed by 7s and 6dπ
orbitals. Four singly occupied orbitals form the additional
bonding. In addition, there are two localized 5f electrons; thus,
in total there are six unpaired electrons with parallel spins and
maximum angular momentum, resulting in a septet state with a
Λ value of 11. Therefore, both 7s, 6d, and 5f atomic orbitals
participate in the bonding in the uranium diatom. For the earlier
actinides we may expect less contribution from 5f orbitals, and
the bonding will instead be dominated by 6d and 7s. Actually,
for Ac2 the 7p orbitals will also play a prominent role in the
bond formation.
2 The Theoretical Approach
The results presented below have been obtained using multiconfigurational quantum chemical methods. The complete active space
(CAS) SCF method8 is used to generate wave functions for a
predetermined set of electronic states. Dynamic correlation energy is
added using second-order perturbation theory, CASPT2.9,10 Relativistic
effects are included based on the Douglas-Kroll-Hess (DKH) Hamiltonian.11,12 The scalar part of this Hamiltonian is used in the generation
of the CASSCF/CASPT2 wave functions. Spin-orbit (SO) effects are
included using an effective one-electron spin-orbit Hamiltonian based
on an atomic mean field approximation of the two-electron part.13 All
CASSCF wave functions of the appropriate symmetries are used as
(8) Roos, B. O. In AdVances in Chemical Physics; Ab Initio Methods in
Quantum Chemistry - II; Lawley, K. P., Ed.; John Wiley & Sons Ltd.:
Chichester, England, 1987; p 399.
(9) Andersson, K.; Malmqvist, P.-A° .; Roos, B. O.; Sadlej, A. J.; Wolinski, K.
J. Phys. Chem. 1990, 94, 5483.
(10) Andersson, K.; Malmqvist, P.-A° .; Roos, B. O. J. Chem. Phys. 1992, 96,
1218.
(11) Douglas, N.; Kroll, N. M. Ann. Phys. 1974, 82, 89.
(12) Hess, B. Phys. ReV. A 1986, 33, 3742.
(13) Hess, B. A.; Marian, C. M.; Wahlgren, U.; Gropen, O. Chem. Phys. Lett.
1996, 251, 365.
10.1021/ja066615z CCC: $33.50 © 2006 American Chemical Society
Ac Bond in the Actinide Diatoms Ac2, Th2, Pa2, and U2
basis functions to set up an SO Hamiltonian, where CASPT2 energies
are used in the diagonal elements. This approach has been shown to
work very well in a number of earlier applications (see ref 14 for a
more detailed account of the approach).
The choice of an adequate active space is crucial in the CASSCF/
CASPT2 approach. Here, this choice is especially complicated because
of the large number of atomic orbitals that may potentially participate
to form the chemical bond. It turns out that the bonding pattern is quite
different in the four molecules, and the choice of the active space will
therefore be discussed separately for each case.
The AO basis set used is of the atomic natural orbital type and has
been especially designed to include the effects of the scalar relativistic
terms of the DKH Hamiltonian. A primitive set 27s24p18d14f 6g3h
was contracted to 9s8p6d5f 2g1h.15 All calculations were performed
with the computer software MOLCAS-6.16 The program VIBROT in
MOLCAS was used to compute spectroscopic constants through
numerical integration of the rovibrational Schrödinger equation.
One quantity that will be used in the discussion is the effectiVe bond
order, EBO. It is defined as follows: Each bonding orbital i has a
natural orbital occupation number bi. The corresponding antibonding
orbital has the occupation number abi. The EBO is then defined as:
EBO )
∑
i
ARTICLES
Figure 1. CASPT2 potentials for the lowest electronic states of Ac2. All
states with a binding energy larger than 1 eV are shown. Labels: A ) B
) 3Σg-, C ) 3Πu, D ) 3Σu+, E ) 5Σu-.
Table 1. Calculated Spectroscopic Constants for the Ground
State of Ac2
(bi - abi)
2
This definition allows for fractional bond orders. If bi is close to two
and abi close to zero the contribution to EBO is one corresponding to
the creation of a complete bond. If, on the other hand both bi and abi
are close to one, no bond is formed. The two electrons are then localized
one on each atom. This happens ,for example, in the U diatom.
Intermediate cases also exist in weakly bonded molecules as we shall
see below. It is thus not certain that what would formally be called a
quintuple bond really has five fully developed bonds. Some of them
may be quite weak, and the EBO may be appreciably smaller than
five. The natural orbital occupation numbers that are used to define
the EBO are very stable quantities with respect to variations in the
basis set and the quality of the wave function once the most important
active orbitals are included. It is therefore a good and stable measure
of bond order.
3 Results
3.1. The Actinium Diatom. The actinium atom has the
electronic ground state (6d )1(7s)2, 2D3/2. One would not expect
this state to be able to form a chemical bond, due to the repulsion
between the closed 7s shells, which hinders the 6d orbital to
get close enough to interact. However, as we shall see below,
this becomes possible through 7s7p hybridization, a unique
feature of the Ac-Ac bond. A triple bond can in principle be
formed from the valence excited states (6d )2(7s)1, 4F3/2, which
are located 1.14 eV above the ground level, but the bond energy
would have to be larger than 2.28 eV to overcome the necessary
valence excitation energy. Another possibility is a mixed
dissociation limit with one atom in the ground state and the
other in the valence excited state, which requires a promotion
energy of 1.14 eV and can in principle give a bond of the order
2.5. Only explicit calculations can determine which choice is
energetically most favorable. The natural choice of active
orbitals for such a study are the 6d and 7s orbitals of both atoms,
(14) Roos, B. O.; Malmqvist, P.-A° . Phys. Chem. Chem. Phys. 2004, 6, 2919.
(15) Roos, B. O.; Lindh, R.; Malmqvist, P.-A° .; Veryazov, V.; Widmark, P. O.
Chem. Phys. Lett. 2005, 295-299, 409.
(16) Karlström, G.; Lindh, R.; Malmqvist, P.-A° .; Roos, B. O.; Ryde, U.;
Veryazov, V.; Widmark, P.-O.; Cossi, M.; Schimmelpfennig, B.; Neogrady,
P.; Seijo, L. Comput. Mater. Sci. 2003, 28, 222.
Re (Å)
3∑ g
5∑ u
3∑ +
u
3∑ -(0 +)
g
g
3∑ -(1 )
g
g
5∑ -(2 )
u
g
5∑ -(1 )
u
g
5∑ -(0 -)
u
u
3∑ +(0 -)
u
g
3∑ +(1 )
u
g
∆G1/2 (cm-1)
D0 (eV)
Te (cm-1)
Results without Spin-Orbit Coupling
3.635
1.336
89
3.391
1.328
102
3.373
1.081
128
56
2243
Results with Spin-Orbit Coupling
3.631
1.185
92
3.633
1.184
88
3.392
1.166
95
3.393
1.165
98
3.393
1.165
99
3.751
0.923
84
3.371
0.908
126
135
151
165
170
2112
2234
in total 12 orbitals with 6 active electrons. Such calculations
were performed but were not successful. The potential curves
were not continuous. At certain distances the nature of the active
orbitals changed, resulting in abrupt changes of the energy.
Inspection showed that the 7p orbitals played an important role,
and it was decided to include them in the active space, which
then increased to 18 orbitals. The resulting potential curves were
now perfectly smooth. They are shown in Figure 1 for a number
of the lowest electronic states (so far without the inclusion of
spin-orbit coupling).
In all, 12 singlet, 11 triplet, and 11 quintet potential curves
were computed. The density of states is high, but two states,
3∑ - and 5∑ -, are well separated and are both candidates for
g
u
the ground state of Ac2. The 3∑g- state has a bond length of
3.64 Å and a bond energy (D0) of 1.34 eV. This state dissociates
to two ground state atoms. The 5∑u- state has a bond length of
3.39 Å and a bond energy of 1.33 eV. This state dissociates to
the mixed limit. The computed energy separation at infinite
distance is 1.13 eV, 0.05 eV less than the experimental value.
The two states are thus degenerate at this level of theory. More
details are given in Table 1.
Inclusion of spin-orbit coupling reduces the bond energy
for the 3∑g- state to 1.18 eV but has very small effects on other
spectroscopic parameters. The reason is that the spin-orbit
coupling is small around equilibrium. Spin-orbit effects will
stabilize the Ac atom by 0.166 eV (assuming the Landé interval
J. AM. CHEM. SOC.
9
VOL. 128, NO. 51, 2006 17001
Roos et al.
ARTICLES
Figure 2. Four strongly occupied molecular orbitals in Ac2. Labels and occupation numbers are given in the figure.
rule to be valid).17 A recent study of the atom using the present
theoretical approach gave the same result.15 However, here it
has not been possible to include all electronic states that would
be needed for a complete treatment of the spin-orbit coupling,
and the SO stabilization at infinite distance is only 0.170 eV,
half of the experimental value for two Ac atoms. It is likely
that the error is much smaller at equilibrium, and we therefore
estimate the true binding energy of Ac2 to be about 1.0 eV.
This value is too small to allow a triple bond to be formed.
Thus, we expect a single bond only. However, the appearance
of the 7p orbitals gives another possibility. The dominant
electronic configuration of the ground state is: (7sσg)2(7s7pσu)2(6dπu)2 with the following Mulliken populations:
7s1.527p0.466d 0.955f 0.04. The bonding orbitals 7sσg and 6dπu give
rise to a double bond with EBO ) 1.72. The 7s7pσu orbital is
an sp hybrid, which yields a lone pair orbital with only little
antibonding character. Thus, the chemical bond in Ac2 is
effectively a double bond. We show the orbitals in Figure 2.
The lowest excited state, 5∑u- and also the next state, 3∑u-,
both have the leading configuration (7sσg)2(7s7pσu)1(6dπu)2(6dσg)1 but with different spin coupling in the 6dπu shell. The
energy difference between the minima of the potential curves,
Te, are 56 and 2243 cm-1, respectively. One more binding orbital
results in a shorter bond distance of 3.37-3.39 Å and an EBO
of 2.5. The results with spin-orbit coupling show five levels
within an energy range of less than 200 cm-1. It is clear that
the accuracy of the calculations does not allow an unambigous
determination of the lowest level.
3.2. The Thorium Diatom. The thorium atom has the
electronic ground level (6d )2(7s)2, 3F2, but the (6d )3(7s)1, 5F1,
level is located only 0.69 eV above the ground state. It is thus
not unlikely that a quadruple bond can be formed between the
two atoms. Such a bond would have the 7sσg and 6dπu orbitals
doubly occupied and the remaining two electrons occupying
orbitals (mainly 6d in character) of other symmetries. On the
basis of this reasoning, an active space of 12 orbitals was chosen,
characterized as 6d and 7s on both atoms. This specifies mainly
the symmetries of the active orbitals. The MCSCF procedure
will determine the exact mixture of atomic shells (6d, 7s, 7p,
and 5f ). It is, however, unlikely that the 5f orbitals will
contribute. They are not occupied in the ground state atoms,
and they form much weaker bonds than the 6d and 7s orbitals.
What about the 7p orbitals that were so important for the
actinium diatom? They are much less important here with a
total population of only about 0.1 electrons in the two lowest
electronic states. There is thus no need to include them in the
active space.
(17) Sansonetti, J.; Martin, W.; Young, S. Handbook of Basic Atomic Spectroscopic Data (Version 1.00). [Online] Available: http://physics.nist.gov/
handbook; National Institute of Standards and Technology: Gaithersburg,
MD, 2003.
17002 J. AM. CHEM. SOC.
9
VOL. 128, NO. 51, 2006
Figure 3. CASPT2 potentials for the lowest electronic states of Th2.
Labels: A ) 3∆g, B ) 3Σg-, C ) 1Σg+, D ) 1∆g, E ) 3Σu+, F ) 3∆u,
G ) 3Φu, H ) 3Πu, I ) 1Φg, J ) 3Πg, K ) 3Φg.
Table 2. Electronic Configurations and Spectroscopic Constants
for Th2 at the CASPT2 Level of Theory
state
3∆
g
1∑ +
g
3∑ +
u
1∆
g
3Φ
u
3Π
u
1Φ
g
1Φ
u
3∑ g
3∆
u
3Π
g
1∆
u
3Φ
g
configuration
Re (Å)
D0 (eV)
∆G1/2 (cm-1)
(7sσg)2(6dδg)(6dσg)(6dπu)4
(7sσg)2(6dσg)2(6dπu)4
(7sσg)2(6dσg)(6dπu)4(7sσu)
(7sσg)2(6dδg)(6dσg)(6dπu)4
(7sσg)2(6dδg)(6dσg)2(6dπu)3
(7sσg)2(6dδg)(6dσg)2(6dπu)3
(7sσg)2(6dσg)2(6dπu)3(7sσu)
(7sσg)2(6dδg)(6dσg)2(6dπu)3
(7sσg)2(6dδg)2(6dσg)2(6dπu)2
(7sσg)2(6dδg)(6dπu)4(7sσu)
(7sσg)2(6dσg)2(6dπu)3(7sσu)
(7sσg)2(6dδg)(6dπu)4(7sσu)
(7sσg)2(6dσg)2(6dπu)3(7sσu)
2.72
2.78
2.90
2.68
2.80
2.80
2.92
2.82
2.68
2.89
2.92
2.72
2.84
3.98
3.93
3.79
3.74
3.48
3.42
3.37
3.28
3.27
3.20
3.17
3.01
2.96
183
186
185
182
173
180
205
167
139
149
172
168
117
CASPT2 potentials were generated for 22 singlet and 23
triplet states. As the lower electronic states were found to have
gerade symmetry, we performed spin-orbit calculations only
for this symmetry. Twenty-three electronic states of gerade
symmetry were used as the basis for the spin-orbit Hamiltonian.
The resulting CASPT2 potentials are shown in Figure 3. The
figure shows a very congested set of energy levels with 12
electronic states within an energy range of less than 1 eV. The
calculations give 3∆g as the ground state but Te to the 1∑g+
state is only 400 cm-1. It is thus impossible from these
calculations to give a definite answer concerning the ground
state of Th2. The spectroscopic constants for the lower electronic
states are presented in Table 2. Both candidates for the ground
state gives the same bonding pattern. A quadruple bond is
formed with six of the eight electrons in the strongly bonding
orbitals 7sσg and 6dπu. The two remaining electrons are
distributed in 6d-based bonding molecular orbitals: the 1∑g+
Ac Bond in the Actinide Diatoms Ac2, Th2, Pa2, and U2
ARTICLES
Table 4. Electronic Configurations and Spectroscopic Constants
for the Lowest Gerade Levels of Pa2 at the CASPT2 Level of
Theory
state
configuration
Re
(Å)
D0
(eV)
∆G1/2
(cm-1)
3∑ g
1Γ
g
5∑ +
g
5∑ +
u
3∆
g
(7sσg)2(6dπu)4(6dδg)2(6dσg)2
(7sσg)2(6dπu)4(6dδg)2(6dσg)2
(7sσg)2(6dπu)4(6dδg)2(6dσg)1(5fσg)1
(7sσg)2(6dπu)4(6dδg)2(5fσg)1(7sσu)1
(7sσg)2(6dπu)4(6dδg)3(6dσg)1
2.37
2.38
2.38
2.39
2.44
5.19
4.99
4.90
4.78
4.65
383
326
295
330
159
Figure 4. Molecular orbitals forming the quadruple bond in Th2. Labels
and occupation numbers are given in the figure.
Table 3. Lowest Energy Levels for Th2 with Spin-Orbit Coupling
state
Ω
Re (Å)
D0 (eV)
state
Ω
Re (Å)
D0 (eV)
3∆
g
1∑ +
g
3∆
g
3∆
g
1
0
2
3
2.76
2.78
2.74
2.75
3.28
3.08
3.22
3.00
1∆
g
3∑ g
3∑ g
2
0
1
2.72
2.73
2.73
2.82
2.64
2.42
state has 6dσg doubly occupied, while one electron is moved to
6dδg in the3∆g state. In both cases a quadruple bond is formed.
EBO is 3.67 in the3∆g state.
The natural orbital occupation numbers for the strongly
occupied orbitals for the 3∆g state are (7sσg)1.89(6dπu)3.82(6dδg)0.97(6dσg)0.95. The quadruple bond is thus fully developed
with only a small population of the antibonding orbitals. The
situation is similar for the 1∑g+ state. The bonding orbitals are
shown in Figure 4. The strength of the quadruple bond manifests
itself in a surprisingly large bond energy, which is close to 4
eV at the CASPT2 level of theory for the lowest two electronic
states (cf. Table 2). Th2 is thus a much more stable molecule
than Ac2.
Adding spin-orbit coupling to these results stabilizes the 3∆g
state by 0.14 eV at equilibrium, while the 1∑g+ state is hardly
affected at all. Calculation on the separate atoms gives an SO
stabilization of 0.84 eV at the dissociation limit. The dissociation
energy is thus reduced by 0.70 eV for the triplet state and by
0.84 eV for the singlet state. We conclude that 3∆g (1g) is the
ground state of Th2 with a bond distance of 2.76 Å and a bond
energy of 3.28 eV. More details are given in Table 3.
3.3. The Protactinium Diatom. The protactinium atom has
the ground level (5f )2(6d )(7s)2, 4K11/2. The J-averaged energy
is 0.77 eV higher. The first known level with 7s singly occupied
is (5f )2(6d )2(7s), 6L11/2, located 0.87 eV above the ground level.
Thus, the valence excitation needed to form a quintuple bond
between two Pa atoms is 1.74 eV, slightly higher than the energy
needed for the quadruple bond in Th2.
We can expect, based on the results obtained for Th2, that a
strong triple bond is formed from the orbitals 7sσg and 6dπu.
The remaining bonds are most likely one-electron bonds
involving 6dσ and 6dδ orbitals with involvement of 5f. We have
based the selection of active orbitals on this assumption and
allowed the distribution of the ten valence electrons among the
16 orbitals of σ, π, and δ character.
Figure 5. CASPT2 potentials for the lowest electronic states of Pa2.
Labels: A ) 3Σg-, B ) 1Σg+, C ) 3Γg, D ) 5Σg+, E ) 5Σu+.
The results for the CASPT2 calculations are shown in Table
4. These results shows that the choice of active space was
adequate. A strong quintuple bond is formed. Three two-electron
bonds ((7sσg)2(6dπu)4) and in addition two one-electron bonds
are formed from the 6dδg orbitals, and one more two-electron
bond is formed that involves a hybrid between 6dσg and 5fσg.
The double occupancy of the 6dδg shell gives rise to three
electronic states, 3∑g-, 1Γg, and 1∑g+. The lowest state is 3∑gwith a binding energy of 5.2 eV and a bond distance of 2.37 A.
The two other states are 0.20-0.25 eV higher. The CASPT2
potentials for the lower electronic states are shown in Figure 5,
and the most strongly occupied molecular orbitals are shown
in Figure 6.
The occupancy of of the binding MOs are very similar in
the three lower states: (7sσg)1.72(6dπu)3.82(6dσg)1.92(6dδg)2.00.
Remaining electrons occupy antibonding orbitals. This yields
an EBO of 4.51 for all three lowest electronic states. The
protactinium diatom thus exhibits a well-developed and strong
quintuple bond. This is the highest bond order and bond energy
computed for the diatoms studied here. The bond energy will
be reduced by the inclusion of spin-orbit coupling but will still
be larger than those for Th2 and U2.
Adding spin-orbit coupling will split the 3∑g- state into 0g+
and 1g components with the latter lowest in energy. The energies
for the five lowest levels are presented in Table 5. We notice
that the bond distance for the 3∑g--derived levels has not
changed, while the bond energy is reduced by 1.2 eV. The SO
stabilization of the 0g+ level is only 0.22 eV, while it is 1.41
eV for the free atoms, which should be compared to the
experimental value 1.54 eV. Again, there is a problem with the
balance between the atomic and molecular values of this
quantity. Both are most certainly underestimated in the calculaJ. AM. CHEM. SOC.
9
VOL. 128, NO. 51, 2006 17003
Roos et al.
ARTICLES
Figure 6. Molecular orbitals forming the quintuple bond in Pa2. Labels and occupation numbers are given in the figure.
Table 5. Lowest Energy Levels for Pa2 with Spin-Orbit Coupling
state
Ω
Re (Å)
D0 (eV)
state
Ω
Re (Å)
D0 (eV)
3∑ g
3∑ g
1Γ
g
0
1
4
2.37
2.37
2.38
4.00
3.79
3.55
5∑ g
3∆
g
2
3
2.40
2.45
3.51
3.41
tions, and the relative error could well be 0.1-0.2 eV. Therefore,
our estimate of the dissociation energy is not very accurate.
We can, however, conclude that Pa2 is the most strongly bound
among the molecules studied here. In the next section we shall
show that the uranium diatom has a much lower bond energy
and a less developed quintuple bond. Extrapolation to the heavier
diatoms becomes possible, and we can conclude that Pa2 is the
most stable diactinide. Even so, it is most unlikely that it will
ever be made because of its high toxicity.
3.4. The Uranium Diatom. Our original study of the U21
showed that the ground state of U2 has a very complex electronic
structure. We have here extended the study of U2 by considering
several electronic states of different spin multiplicities, namely
38 septet states lying up to 0.36 eV (2930 cm-1) and 32 quintet
states lying up to 0.78 eV (6290 cm-1) above the ground state.
These states have been allowed to interact via the spin-orbit
coupling Hamiltonian, resulting in 386 spin-orbit states. For
all these states we have computed some points around the
equilibrium bond distance.
The uranium atom has the ground-state electronic configuration (5f )2(6d )1(7s)2, 5L6. In the U2 molecule each uranium
atom has in principle six electrons available for forming
chemical bonds. Trial studies including different sets of valence
orbitals in the active space showed that a strong triple bond
was formed involving the 7sσg and 6dπu orbitals. The occupation
numbers of these three orbitals were close to two with small
occupation of the corresponding antibonding orbitals. It was
therefore decided to leave these orbitals outside the active space
and also to remove the antibonding orbitals 7sσu and 6dπg. This
approximation should work well around equilibrium but of
course prevents the calculation of full potential curves. The
remaining six electrons occupy the remaining 5f and 6d orbitals,
resulting in a final active space of six electrons in twenty orbitals
(6/20). The most stable electronic state is a septet state (where
the six electrons have parallel spins) and has a total orbital
angular momentum (Λ) value of 11. A quintuple bond connects
17004 J. AM. CHEM. SOC.
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the two atoms. It is built from three electron pair bonds formed
by 7s and 6dπ orbitals, and four singly occupied orbitals form
the additional bonding. The total EBO is 4.2 because the
bonding involving 5f atomic orbitals is weak with large
occupations of the antibonding orbitals. In addition there are
two localized 5f electrons. The spin-orbit calculations showed
that the spin and orbital angular momenta combine to form a
Ω ) 8 state. The final label of the ground state is thus 7O8.
The computed bond energy for this state is, however, only 1.15
eV, and a second state with Ω ) 8 (Λ ) 12) lies only 0.01 eV
higher in energy and has a bond distance of 2.46 Å. It is thus
impossible from this study to uniquely determine the ground
state of U2. It should also be emphasized that the determination
of the bond energy is uncertain due to the difficulty of achieving
a balanced description of the spin-orbit stabilization of the
energy. The bond energy was computed using the total energy
of two uranium atoms obtained with an active space comprising
the necessary 7s, 6d, and 5f orbitals, which is not completely
balanced compared to the active space used for the molecule.
The main terms of the multiconfigurational wave function
are:
ψ (S ) 3, Λ ) 11) ) 0.782(7sσg)2(6dπu)4(6dσg)(6dδg)
(5fπu)(5fφu)(5fφg) + 0.596(7sσg)2(6dπu)4(6dσg)(6dδg)
(5fδu)(5fπg)(5fφu)(5fφg)
The energy levels around equilibrium are incredibly dense. With
the inclusion of spin-orbit coupling 64 states lie within 1 eV
above the ground state. Limited portions of the potential energy
curves with the inclusion of spin-orbit coupling, for the lowest
states are reported in Figure 7. The equilibrium bond distances
and the spin-free composition of these states are reported in
Table 6.
4 Summary and Discussion
Above we have presented the results of the quantum chemical
calculations for each of the diatoms separately. Here we shall
summarize these results and compare them to see the trends
and possibly make some extrapolations to consider the possibility of forming diatoms of the heavier actinides.
Let us first find the common elements in the electronic
structure. We present the main electronic configurations in Table
Ac Bond in the Actinide Diatoms Ac2, Th2, Pa2, and U2
ARTICLES
Figure 7. CASPT2-SO potential energy curves for the lowest electronic
states of U2.
Table 6. Lowest Gerade Energy Levels for U2 with the Inclusion
of Spin-Orbit Coupling
Λ
Ω
Re
(Å)
11
12
8
9
2.43
2.46
D0
(eV)
ωe
(cm-1)
Λ
1.15
1.14
331
389
11
10
Ω
Re
(Å)
D0
(eV)
ωe
(cm-1)
8
7
2.44
2.43
1.00
0.93
293
292
Table 7. Electronic Structure of the Early Diactinides
Ac2
Th2
Pa2
U2
(7sσg)2(7s7pσu)2(6dπu)2
(7sσg)2(6dπu)4(6dδg)1(6dσg)1
(7sσg)2(6dπu)4(6dδg)2(5f6dσg)2
(7sσg)2(6dπu)4(6dσg)1(6dδg)1(5fδg)1(5fπu)1(5fφu)1(5fφg)1
3∑ g
3∆
g
3∑ g
7Ο
g
Table 8. Mulliken Populations (per atom), Bond Distances (Re),
and Bond Energies (D0) for the Early Diactinides
Ac2
Th2
Pa2
U2
7s
7p
6d
5f
Re (Å)
D0 (eV)
1.49
0.93
0.88
0.94
0.49
0.01
0.02
0.00
0.96
2.83
3.01
2.59
0.04
0.21
1.06
2.44
3.63
2.76
2.37
2.43
1.2
3.3
4.0
1.2
7. The main bonding motif in all of these elements is represented
by the orbitals 7sσg and 6dπu. They are the strongest bonding
orbitals and are completely filled in all diatoms except Ac2
where the atomic promotion energy is to large to allow a
complete decoupling of the 7s orbitals. Instead a lone-pair orbital
is formed through 7s,7p hybridization. The remaining electrons
reside in 6d,5f hybrid orbitals, which form weaker one-electron
bonds (except in Pa2 where a second σg orbital is doubly
occupied).
Table 8 and Figure 8 give the Mulliken populations on each
atom. Some trends emerge from these numbers: the 7s orbital
is singly occupied, except in Ac2 as noted above. Only in this
diatom is there any appreciable contribution from the 7p orbital.
The 6d population increases up to Pa2 and then starts to decrease,
while the population of 5f continuously increases, as expected.
It is small for Ac and Th, which are more similar to the
corresponding transition metal series. Up to Pa2 the more
strongly bonding 6d orbitals can contribute to the bonding
orbitals, even if there is some hybridization with 5f. For U2
this is no longer possible. The atomic promotion energy is too
large. As a result, the electronic structure retains some of its
atomic character with one nonbonding electron on each atom
and two only weakly bonding 5f-generated MOs.
Figure 8. Populations of the different atomic shells (7s,6p,6d,5f ) in the
atoms of the actinide diatoms.
The bond in U2 is consequently weaker than that of Pa2. Table
8 also presents the bond distances and the bond energies. The
latter have rather large uncertainties (due to the difficulty of
achieving a balanced treatment of spin-orbit coupling), but the
trends are clear. The bonds get shorter and stronger up to Pa2,
and then the trends reverse for U2 where the 5f to 6d promotion
becomes less effective. Only Th2 and Pa2 are able to use all
electrons effectively to create a quadruple and quintuple bond,
respectively.
What about the heavier actinides? The next is neptunium,
which has a (5f )4(6d )1(7s)2 ground state. The first level with
more unpaired electrons is (5f )4(6d )2(7s)1, 0.88 eV above the
ground level. A triple bond would thus have to overcome an
atomic promotion energy of 1.76 eV, which might be possible
with some help of the weakly bonding 5f electrons, but the bond
will certainly be weak. Plutonium has the ground-state configuration (5f )6(7s)2 with no 6d orbitals occupied. A (5f )5(6d )1(7s)2 level is located 0.78 eV above the ground level, but 2.58
eV is needed to unpair the 7s orbital. Only a weak bond can
possibly be formed in this case. The promotion energies then
increase further for the heavier elements. At the same time the
5f orbitals become more and more corelike and unlikely to
participate in the bonding. We thus do not expect any of the
heavier actinides to form diatoms with bond energies as large
as those obtained here.
We are then in the position to summarize the properties of
the actinide diatoms. The strongest bond is found for Pa2 with
the bond for Th2 slightly weaker. It is possible that the Np
diatom is also stable, but this is probably the end of the story.
The crucial features in the bond formation are the access to a
singly occupied 7s orbital on each atom and low promotion
energies from 5f to 6d.
It is interesting to extend the view to the rest of the periodic
table. Quintuply bonded molecules also exist among the
transition metals and their ions. The molecule ArCrCrAr was
recently synthesized.18 The Cr(I)-Cr(I) unit has a short bond
distance (1.8 Å), and subsequent quantum chemical calculations
showed that a weak quintuple bond is formed with a bond
(18) Nguyen, T.; Sutton, A. D.; Brynda, M.; Fettinger, J. C.; Long, G. J.; Power,
P. P. Science 2005, 310, 844.
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Roos et al.
ARTICLES
energy of about 3 eV.19 More strongly quintuply bound diatoms
of transition metals may exist. A strong candidate is Nb2 because
Nb has a 4d 4s1 ground state in contrast to V and Ta. Earlier
theoretical studies seem to confirm that this is the case.20
The highest bond order found in the periodic table is the
hextuply bound W2 molecule with a bond energy of about 5
eV.21 It has an EBO of 5.2. No other diatom has a higher bond
(19) Brynda, M.; Gagliardi, L.; Widmark, P.-O.; Power, P. P.; Roos, B. O.
Angew. Chem., Int. Ed. 2006, 45, 3804.
(20) James, A. M.; Kowalczyk, P.; Fournier, R.; Simard, B. J. Chem. Phys. 99,
8504.
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VOL. 128, NO. 51, 2006
order. It is interesting to note that one of the diatoms that comes
closest to this value is the Pa2 molecule with EBO ) 4.5.
Acknowledgment. This work was supported by the Swedish
Natural Science Research council (VR) and the Swedish
Foundation for Strategic Research (SSF). L.G. thanks the Swiss
National Science Foundation Grant 200021-111645/1 for support.
JA066615Z
(21) Roos, B. O.; Borin, A. C.; Gagliardi, L. Angew. Chem., Int. Ed. 2006, in
press.